Question: Factor the following expression: $-8$ $x^2+$ $31$ $x+$ $45$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(45)} &=& -360 \\ {a} + {b} &=& & & {31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-360$ and add them together. Remember, since $-360$ is negative, one of the factors must be negative. The factors that add up to ${31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${40}$ $ \begin{eqnarray} {ab} &=& ({-9})({40}) &=& -360 \\ {a} + {b} &=& {-9} + {40} &=& 31 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-8}x^2 {-9}x +{40}x +{45} $ Group the terms so that there is a common factor in each group: $ ({-8}x^2 {-9}x) + ({40}x +{45}) $ Factor out the common factors: $ x(-8x - 9) - 5(-8x - 9) $ Notice how $(-8x - 9)$ has become a common factor. Factor this out to find the answer. $(-8x - 9)(x - 5)$